Latent Roots and Matrix Variates: A Review of Some Asymptotic Results

The exact noncentral distributions of matrix variates and latent roots derived from normal samples involve hypergeometric functions of matrix argument. These functions can be defined as power series, by integral representations, or as solutions of differential equations, and there is no doubt that these mathematical characterizations have been a unifying influence in multivariate noncentral distribution theory, at least from an analytic point of view. From a computational and inference point of view, however, the hypergeometric functions are themselves of very limited value due primarily to the many difficulties involved in evaluating them numerically and consequently in studying the effects of population parameters on the distributions. Asymptotic results for large sample sizes or large population latent roots have so far proved to be much more useful for such problems. The purpose of this paper is to review some of the recent results obtained in these areas.